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‘Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously.’
D. L. Goodstein, States of Matter.
The goal of statistical mechanics is to consider the phenomena of continuum mechanics based on the motion of atoms and molecules. The theory originated in the second half of the 19th century as kinetic theory; the term statistical mechanics was coined by Gibbs in his book ‘Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics‘ in 1902. This level of consideration is now referred to as classical statistical mechanics. However, the development of kinetic theory in the 19th century already showed that classical mechanics was unsuitable for describing phenomena at the atomic and molecular level. Below is an expressive quote from Gibbs, which clearly demonstrates the state of physics in the late 19th century (during the writing of this book):
‘In the present state of science, it seems hardly possible to frame a dynamic theory of molecular action which shall embrace the phenomena of thermodynamics, of radiation, and of the electrical manifestations which accompany the union of atoms. Yet any theory is obviously inadequate which does not take account of all these phenomena. Even if we confine our attention to the phenomena distinctively thermodynamic, we do not escape difficulties in as simple a matter as the number of degrees of freedom of a diatomic gas. It is well known that while theory would assign to the gas six degrees of freedom per molecule, in our experiments on specific heat we cannot account for more than five. Certainly, one is building on an insecure foundation, who rests his work on hypotheses concerning the constitution of matter.’
Quantum mechanics that appeared in the first half of the 20th century led to the development of quantum statistical mechanics, which resolved the problems of classical statistical mechanics. However, the conceptual framework of quantum statistical mechanics relies on the wave function of the entire system. This complicates the visual representation of the system, and on the other hand, it raises the question of what a wave function is, leading to discussions about the interpretation of quantum mechanics.
The quasi-classical approximation of statistical mechanics allows us to partly return to the visual image of classical statistical mechanics by means of quantum mechanical corrections. It is important to recognize that the atomism of the 19th century belongs to the past, but nevertheless, classical statistical mechanics with necessary corrections can be used to solve problems. This approach puts the interpretation of quantum mechanics to background and instead leads to a hierarchy of conceptual models, which we will explore in the first chapter.
The processes for candle combustion, both in classical and quantum statistical mechanics, lead to a system of equations that cannot even be written down. Therefore, it is impossible in principle to find a solution to such a system of equations, but one can mathematically prove certain theorems that have practical significance. For example, the equipartition theorem proven in the 19th century led to discrepancies between the heat capacities of gases predicted by the theorem and the experimental values. Thus, the general properties of the system of equations of classical statistical mechanics led to the conclusion of the inadequacy of classical mechanics for molecular motion.
The way forward was possible by using distribution functions. Equilibrium statistical mechanics is based on equilibrium distributions and this allows us in a number of cases to estimate the thermodynamic properties of substances from molecular constants. This development in considered in the second chapter for an ideal molecular gas. For example, the heat capacities and entropies of gases (CO2, CO, O2) used in Chapter 1.5 ‘Adiabatic Flame Temperature‘ were obtained in this way.
The success of equilibrium statistical mechanics is connected to the calculation of substance macroscopic properties, which are then used to solve the equations of continuum mechanics. In other words, this involves the combined use of theories at different levels of organization. Statistical mechanics allows us to estimate thermodynamic properties, which are then used at the level of classical thermodynamics.
At the same time, in philosophy of physics, classical thermodynamics is assumed to be reducible to statistical mechanics. This topic is examined in more detail in the next part of the book, in this part we focus on the fate of the Clausius inequality in statistical mechanics, as the Clausius inequality is associated with the arrow of time. Ludwig Boltzmann provided a statistical justification for the arrow of time, but Boltzmann’s analysis was limited to the case of the ideal monatomic gas. Gibbs extended entropy in statistical mechanics to the general case, but due to the peculiarities of the phase space formalism, the Gibbs statistical entropy remains constant during an irreversible process. We consider this tragic story in Chapter 3.
The problems with the arrow of time in statistical mechanics coincided with the arrival of Shannon’s information theory. Shannon introduced the concept of information entropy, which has a similar mathematical structure as compared with the Gibbs statistical entropy. This led to two contradictory desires: on the one hand, physicists wanted to find a physical basis for information, on the other, to connect entropy with subjectivity. As an example of attempts to combine information and thermodynamic entropy, below there are several quotes from Semikhatov’s popular science book, from the chapter with the expressive title ‘Shredding into Ignorance‘:
‘Entropy is a measure of ignorance of details.’
‘The ignorance to which we are condemned turns out not to be a philosophical category only, but it is also expressed by a quantity that in principle can be measured. This quantity is called entropy: the greater the entropy, the greater the ignorance (the more molecules can do things that completely escape our attention).’
‘The solution was proposed by Boltzmann, and the formula expressing entropy “via molecules” is now engraved on his tombstone. According to this formula, entropy measures the extent of our ignorance about the precise movement of molecules.’
‘The equilibrium state is the state of maximum ignorance.’
‘The idea of using knowledge about the behavior of molecules to perform work involves the laws of information processing, which at the most fundamental level are related to the laws of molecular motion.’
The last chapter discusses the current movement in statistical mechanics for the entropy objectivity under the slogan ‘Back to Boltzmann’. This allow us to discuss non-equilibrium states in statistical mechanics in more detail and compare them with those in continuum mechanics. Finally, we examine the practical applications of non-equilibrium statistical mechanics; this shows that conceptual issues with the arrow of time do not prevent physicists from finding the correct kinetic equations.
Next: Chapter 1. Conceptual Models of Statistical Mechanics
References
A. Semikhatov, Everything That Moves. Walk over the Restless Universe from Space Orbits to Quantum Fields (In Russian), 2022, Walk 9 ‘Shredding into Ignorance‘.
Discussion
https://evgeniirudnyi.livejournal.com/403279.html
23.02.2025 Entropy as Ignorance
Semikhatov’s quotes.
https://evgeniirudnyi.livejournal.com/395381.html
22.08.2025 Caution: Statistical Mechanics
Goodstein’s quote.